What Graphs are 2-Dot Product Graphs?

TL;DR

This paper investigates 2-dimensional dot product graphs, exploring their properties and relationships with other graph classes to better understand their structural characteristics and computational complexity.

Contribution

The paper analyzes the position of 2-dot product graphs within the broader landscape of graph classes and their relation to intersection graphs.

Findings

2-dot product graphs have unique structural properties.

Recognition of 2-dot product graphs is computationally complex.

Relationships with other graph classes are characterized.

Abstract

Let $d \geq 1$ be an integer. From a set of $d$-dimensional vectors, we obtain a $d$-\dpg\ by letting each vector $\va^u$ correspond to a vertex $u$ and by adding an edge between two vertices $u$ and $v$ if and only if their dot product $\va^{u} \cdot \va^{v} \geq t$, for some fixed, positive threshold~$t$. Dot product graphs can be used to model social networks. Recognizing a $d$-dot product graph is known to be \NP-hard for all fixed $d\geq 2$. To understand the position of $d$-dot product graphs in the landscape of graph classes, we consider the case $d=2$, and investigate how $2$-dot product graphs relate to a number of other known graph classes including a number of well-known classes of intersection graphs.